{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2022-07-31T02:33:47.231260Z",
     "iopub.status.busy": "2022-07-31T02:33:47.231000Z",
     "iopub.status.idle": "2022-07-31T02:33:47.322890Z",
     "shell.execute_reply": "2022-07-31T02:33:47.322003Z"
    },
    "init_cell": true,
    "origin_pos": 4,
    "slideshow": {
     "slide_type": "notes"
    },
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [],
   "source": [
    "import torch\n",
    "from torch import nn\n",
    "from torch.utils import data\n",
    "import torchvision\n",
    "from torchvision import transforms\n",
    "from torch.utils.tensorboard import SummaryWriter\n",
    "\n",
    "from IPython.core.interactiveshell import InteractiveShell\n",
    "InteractiveShell.ast_node_interactivity = \"all\""
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "<div class=\"jumbotron\">\n",
    "    <p class=\"display-1 h1\">softmax回归的简洁实现</p>\n",
    "    <hr class=\"my-4\">\n",
    "    <p>主讲：李岩</p>\n",
    "    <p>管理学院</p>\n",
    "    <p>liyan@cumtb.edu.cn</p>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "- 通过深度学习框架的高级API能够使实现softmax回归变得更加容易"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- 使用Fashion-MNIST数据集"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- 保持批量大小为256"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# 读取数据"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "batch_size = 256\n",
    "train_iter, test_iter = load_data_fashion_mnist(batch_size)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "def get_dataloader_workers():  \n",
    "    \"\"\"使用2个进程来读取数据\"\"\"\n",
    "    return 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "def load_data_fashion_mnist(batch_size, resize=None):  \n",
    "    \"\"\"下载Fashion-MNIST数据集，然后将其加载到内存中\"\"\"\n",
    "    trans = [transforms.ToTensor()]   # 将图像从PIL类型转换成张量\n",
    "    if resize:\n",
    "        trans.insert(0, transforms.Resize(resize)) # resize是把小图片放大,在0之前插入\n",
    "    trans = transforms.Compose(trans)   # 组合多个transform\n",
    "    mnist_train = torchvision.datasets.FashionMNIST(\n",
    "        root=\"../data\", train=True, transform=trans, download=False)\n",
    "    mnist_test = torchvision.datasets.FashionMNIST(\n",
    "        root=\"../data\", train=False, transform=trans, download=False)\n",
    "    return (data.DataLoader(mnist_train, batch_size, shuffle=True,\n",
    "                            num_workers=get_dataloader_workers()),\n",
    "            data.DataLoader(mnist_test, batch_size, shuffle=False,\n",
    "                            num_workers=get_dataloader_workers()))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# 建立模型，初始化模型参数"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- 需在`Sequential`中添加一个带有10个输出的全连接层"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2022-07-31T02:33:47.326549Z",
     "iopub.status.busy": "2022-07-31T02:33:47.326134Z",
     "iopub.status.idle": "2022-07-31T02:33:47.332207Z",
     "shell.execute_reply": "2022-07-31T02:33:47.331417Z"
    },
    "origin_pos": 7,
    "slideshow": {
     "slide_type": "fragment"
    },
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Sequential(\n",
       "  (0): Flatten(start_dim=1, end_dim=-1)\n",
       "  (1): Linear(in_features=784, out_features=10, bias=True)\n",
       ")"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "net = nn.Sequential(nn.Flatten(), nn.Linear(784, 10))\n",
    "# nn.Flatten()，保留第一个维度的值（batch_size)，把其它维度全部变成一个维度，结果是一个二维数组\n",
    "net"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "\\begin{problem}\\label{prob:flatten}\n",
    "为什么要在nn.Linear()之前添加nn.Flatten()?\n",
    "\\end{problem}\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "> `PyTorch`不会隐式地调整输入的形状，因此，在线性层前定义了展平层（flatten），来调整网络输入的形状"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "- 以均值0和标准差0.01随机初始化权重"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "def init_weights(m):\n",
    "    if type(m) == nn.Linear: # 如果遇到线性层，把该层权重置为normal（0，0.01）\n",
    "        nn.init.normal_(m.weight, std=0.01)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Sequential(\n",
       "  (0): Flatten(start_dim=1, end_dim=-1)\n",
       "  (1): Linear(in_features=784, out_features=10, bias=True)\n",
       ")"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "net.apply(init_weights)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "- ```python\n",
    "torch.nn.init.normal_(tensor, mean = 0, std = 1)\n",
    "```\n",
    "    - 从正态分布采样填充给定的二维矩阵（张量）"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- ```python\n",
    "Moduel.apply(fn)\n",
    "```\n",
    "    - `Module`可以是`torch`创建的模型或者网络\n",
    "    - 对`Module`的每一层应用函数`fn`\n",
    "    - 通常用于初始化模型参数"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# 交叉熵损失函数"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2022-07-31T02:33:47.335522Z",
     "iopub.status.busy": "2022-07-31T02:33:47.334915Z",
     "iopub.status.idle": "2022-07-31T02:33:47.338906Z",
     "shell.execute_reply": "2022-07-31T02:33:47.338142Z"
    },
    "origin_pos": 11,
    "slideshow": {
     "slide_type": "fragment"
    },
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [],
   "source": [
    "loss = nn.CrossEntropyLoss(reduction='none')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- `reduction`：对计算结果的简化方法，`'none'`不做简化，`'mean'`求均值，`'sum'`求和"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "> `CrossEntropyLoss`里面包含了`softmax`和`log`的计算，避免出现数值的上溢（overflow）和下溢（underflow）"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "对于softmax函数\n",
    "\n",
    "$$\\hat y_j = \\frac{\\exp(o_j)}{\\sum_k \\exp(o_k)}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- 如果$o_k$中的一些数值非常大，$\\exp(o_k)$可能大于数据类型容许的最大数字，即**上溢**\n",
    "    - 这将使分母或分子变为`inf`，最后得到0、`inf`或`nan`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- 解决上面问题的技巧：在继续softmax计算之前，先从所有$o_k$中减去$\\max(o_k)$\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\hat y_j & =  \\frac{\\exp(o_j - \\max(o_k))\\exp(\\max(o_k))}{\\sum_k \\exp(o_k - \\max(o_k))\\exp(\\max(o_k))} \\\\\n",
    "& = \\frac{\\exp(o_j - \\max(o_k))}{\\sum_k \\exp(o_k - \\max(o_k))}.\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "- 此时可能有些$o_j - \\max(o_k)$具有较大的负值，$\\exp(o_j - \\max(o_k))$将有接近零的值，即**下溢**\n",
    "    - 这些值可能会四舍五入为零，使$\\hat y_j$为零，并且使得$\\log(\\hat y_j)$的值为`-inf`，反向传播几步后，可能会出现`nan`结果"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- 尽管要计算指数函数，但在计算交叉熵损失时会取它们的对数。因此，通过将softmax和交叉熵结合在一起，可以避免反向传播过程中的数值稳定性问题\n",
    "- 如下面的等式所示，避免计算$\\exp(o_j - \\max(o_k))$，而可以直接使用$o_j - \\max(o_k)$\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\log{(\\hat y_j)} & = \\log\\left( \\frac{\\exp(o_j - \\max(o_k))}{\\sum_k \\exp(o_k - \\max(o_k))}\\right) \\\\\n",
    "& = \\log{(\\exp(o_j - \\max(o_k)))}-\\log{\\left( \\sum_k \\exp(o_k - \\max(o_k)) \\right)} \\\\\n",
    "& = o_j - \\max(o_k) -\\log{\\left( \\sum_k \\exp(o_k - \\max(o_k)) \\right)}.\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- $\\log{\\left( \\sum_k \\exp(o_k - \\max(o_k)) \\right)}$是有界的，可以利用LogSumExp的性质得到"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "- LogSumExp：\n",
    "\n",
    "$$\n",
    "LSE(x_1,\\cdots,x_n)=\\log\\left(\\exp(x_1)+\\cdots+\\exp(x_n)\\right)\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- LogSumExp是对$\\underset{i}{\\max}x_i$的近似\n",
    "\n",
    "$$\n",
    "\\max\\{x_1,\\cdots,x_n\\}\\le LSE(x_1,\\cdots,x_n)\\le \\max\\{x_1,\\cdots,x_n\\}+\\log(n)\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# 优化器"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- 使用学习率为0.1的小批量随机梯度下降作为优化算法"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2022-07-31T02:33:47.341773Z",
     "iopub.status.busy": "2022-07-31T02:33:47.341507Z",
     "iopub.status.idle": "2022-07-31T02:33:47.345445Z",
     "shell.execute_reply": "2022-07-31T02:33:47.344674Z"
    },
    "origin_pos": 15,
    "slideshow": {
     "slide_type": "fragment"
    },
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [],
   "source": [
    "trainer = torch.optim.SGD(net.parameters(), lr=0.1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# 正式训练"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- 调用在softmax_scratch中定义的训练函数`train_ch3`训练模型"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2022-07-31T02:33:47.348296Z",
     "iopub.status.busy": "2022-07-31T02:33:47.348024Z",
     "iopub.status.idle": "2022-07-31T02:34:20.463682Z",
     "shell.execute_reply": "2022-07-31T02:34:20.462914Z"
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    "origin_pos": 18,
    "slideshow": {
     "slide_type": "fragment"
    },
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [
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      "text/plain": [
       "<Figure size 756x540 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from softmax_regression_scratch import train_ch3\n",
    "num_epochs = 10\n",
    "train_ch3(net, train_iter, test_iter, loss, num_epochs, trainer)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "- 如何在`a.ipynb`调用`b.ipynb`文件中已经定义好的函数`fn`？"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- 方法：\n",
    "    - 将`b.ipynb`download as `b.py`文件\n",
    "    - 将`b.py`文件上传到notebook中，与`a.ipynb`同目录\n",
    "    - 在`a.ipynb`中可以用`from b import fn`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# 预测检验集上图像类别"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
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      ],
      "text/plain": [
       "<Figure size 648x108 with 6 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from softmax_regression_scratch import predict_ch3\n",
    "predict_ch3(net,test_iter)"
   ]
  }
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